A perfect number has the property that equals the sum of the divisors of . (For example, 6 is perfect because .) It is not known if there are infinitely many perfect numbers.

Every positive integer can be expressed as the sum of four squares. The numbers to be squared can be positive, negative, or zero and we count different orderings of the summands as different sums. There are some positive integers that cannot be expressed as the sum of three squares; those that can are of the form .

Goldbach conjectured that every even number is the sum of two primes.